The essentially tame local Langlands correspondence, III: the general case

TY - JOUR

T1 - The essentially tame local Langlands correspondence, III

T2 - the general case

AU - Bushnell, Colin J.

AU - Henniart, Guy

PY - 2010/9

Y1 - 2010/9

N2 - Let F be a non-Archimedean local field with finite residue field. An irreducible smooth representation of the Weil group W-F of F is called essentially tame if its restriction to wild inertia is a sum of characters. Let G(n)(et)(F) denote the set of isomorphism classes of irreducible, essentially tame representations of W-F of dimension n. The Langlands correspondence induces a bijection of G(n)(et) (F) with a certain set A(n)(et) (F) of irreducible cuspidal representations of GL(n)(F). We work with the set P-n(F) of isomorphism classes of admissible pairs (E/F, xi) of degree n. There is an obvious bijection of P-n(F) with G(n)(et) (F) and an explicit bijection of P-n(F) with A(n)(et)(F). Together, these maps give a canonical bijection of G(n)(et) (F) with A(n)(et)(F). We showed in an earlier paper that the Langlands correspondence is obtained by composing the map P-n(F) -> A(n)(et)(F) with a permutation of Pn(F) of the form (E/F, xi) bar right arrow (E/F, mu(xi)xi), where mu(xi) is a tamely ramified character of E-x depending on xi. In this paper, we show that there is a canonical choice for the character mu(xi) and determine it explicitly.

AB - Let F be a non-Archimedean local field with finite residue field. An irreducible smooth representation of the Weil group W-F of F is called essentially tame if its restriction to wild inertia is a sum of characters. Let G(n)(et)(F) denote the set of isomorphism classes of irreducible, essentially tame representations of W-F of dimension n. The Langlands correspondence induces a bijection of G(n)(et) (F) with a certain set A(n)(et) (F) of irreducible cuspidal representations of GL(n)(F). We work with the set P-n(F) of isomorphism classes of admissible pairs (E/F, xi) of degree n. There is an obvious bijection of P-n(F) with G(n)(et) (F) and an explicit bijection of P-n(F) with A(n)(et)(F). Together, these maps give a canonical bijection of G(n)(et) (F) with A(n)(et)(F). We showed in an earlier paper that the Langlands correspondence is obtained by composing the map P-n(F) -> A(n)(et)(F) with a permutation of Pn(F) of the form (E/F, xi) bar right arrow (E/F, mu(xi)xi), where mu(xi) is a tamely ramified character of E-x depending on xi. In this paper, we show that there is a canonical choice for the character mu(xi) and determine it explicitly.

U2 - 10.1112/plms/pdp053

DO - 10.1112/plms/pdp053

M3 - Article

VL - 101

SP - 497

EP - 553

JO - PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY

JF - PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY

IS - 2

ER -